Wellbore trajectory models are used for two distinct purposes. The first use is planning the well location, which consists of determining kick-off points, build and drop rates, and straight sections needed to reach a specified target. The second use is to integrate measured inclination and azimuth angles to determine a well's location.
Various trajectory models have been proposed, with varying degrees of smoothness. The simplest model, the tangential model, consists of straight line sections. Thus, the slope of this model is discontinuous at survey points. The most commonly used model is the minimum curvature model, which consists of circular-arc sections. This model has continuous slope, but discontinuous curvature. In fact, the minimum curvature model argues that a wellbore would not necessarily have continuous curvature.
Analysis of drillstring loads is typically done with drillstring computer models. By far the most common method for drillstring analysis is the “torque-drag” model originally described in the Society of Petroleum Engineers article “Torque and Drag in Directional Wells—Prediction and Measurement” by Johancsik, C. A., Dawson, R. and Friesen, D. B., which was later translated into differential equation form as described in the article “Designing Well Paths to Reduce Drag and Torque” by Sheppard, M. C., Wick, C. and Burgess, T. M.
Torque-drag modeling refers to the torque and drag related to drillstring operation. Drag is the excess load compared to rotating drillstring weight, which may be either positive when pulling the drillstring or negative while sliding into the well. This drag force is attributed to friction generated by drillstring contact with the wellbore. When rotating, this same friction will reduce the surface torque transmitted to the bit. Being able to estimate the friction forces is useful when planning a well or analysis afterwards. Because of the simplicity and general availability of the torque-drag model, it has been used extensively for planning and in the field. Field experience indicates that this model generally gives good results for many wells, but sometimes performs poorly.
In the standard torque-drag model, the drillstring trajectory is assumed to be the same as the wellbore trajectory, which is a reasonable assumption considering that surveys are taken within the drillstring. Contact with the wellbore is assumed to be continuous. However, given that the most common method for determining the wellbore trajectory is the minimum curvature method, the wellbore shape is less than ideal because the bending moment is not continuous and smooth at survey points. This problem is dealt with by neglecting bending moment but, as a result of this assumption, some of the contact force is also neglected.
Therefore, there is a need for a new wellbore trajectory model that has sufficient smoothness to model the drillstring trajectory.
There is a further need to provide a new wellbore trajectory model that transforms the simple torque-drag drill string model into a full stiff-string formulation because, in this formulation, drill string bending and shear forces arise that cannot be determined correctly with conventional wellbore trajectory models.